Integral and series representations of q-polynomials and functions: Part I

By applying an integral representation for qk2, we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of q-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include q-Bessel functions, the Ramanujan function, Stieltjes–Wigert polynomials, q-Hermite and q−1-Hermite polynomials, and the q-exponential functions eq, Eq and ℰq. Their representations are in turn used to derive many new identities involving q-functions and polynomials. In this paper, we also present contour integral representations for the above mentioned functions and polynomials.